I would like to find the following service. A website where I can
register myself and submit some text (without any paying). The site
should be able to prove that I said what I said at the time when I
said that. I need this for the following reasons. If somebody (let say
in 2 years) pretends to be inventor of my idea, I can refer to this
site and prove (juridically) that I was the first one who had this
idea. I do not want to get money form this idea. I just want to be
able prove that at that time I already had this idea.

Web site contents are sometimes difficult to date. Why not post
yourself, or a reliable friend or independent person, a sealed letter
with all the information contained therein. Sign across the sealed
down flap. When it arrives the post mark will show date of posting.
Retain the letter to be opened when you need to prove your point.
answerfinder-ga

If it comes to a legal fight, a deposition or affdavit made before a
notary could be a better document:
http://www.dos.state.pa.us/bcel/cwp/view.asp?a=1111&q=432113
http://www.sos.state.wi.us/pdf/notary_info_brochure.pdf

Many states have similar sites.

Sealed letters are useful
- you can also lodge something with a solicitor (lawyer)

You could play with the Way Back machine as belt and braces

http://www.archive.org/web/web.php

Text embedded in a JPEG ( a little sophistication required )
- if you can get your site into their list, then ...

I've just checked one of my sites from 1998 on - a very convincing record

I am not sure that registering 'the idea' will give any protection for
an invention.

If I were you, I would take out a patent.

That's the usual route.

The Answer you are asking for is more commenly used in copyrights, 
Like mentioned above you are better off to get a patent for your work

I do not think that my idea is enough ?scientific? (or even serious)
for patent. I do not want to spend my money and time to make patent. I
just want to do something to make people believe that I was the first
person who had this idea. (Or at leas I want that people believe that
I had this idea already on 2006.)

I've seen A LOT of amazing NEW ideas ....

But when a patent search is done, it turns out that dozens of people
have already been there.

I don't want to discourage you but I would urge caution.